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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Closed convex invariant subsets of $ L\sb{p}(G)$


Author: Anthony To Ming Lau
Journal: Trans. Amer. Math. Soc. 232 (1977), 131-142
MSC: Primary 43A15
DOI: https://doi.org/10.1090/S0002-9947-1977-0477604-5
MathSciNet review: 0477604
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Abstract: Let G be a locally compact group. We characterize in this paper closed convex subsets K of $ {L_p}(G),1 \leqslant p < \infty $, that are invariant under all left or all right translations. We prove, among other things, that $ K = \{ 0\} $ is the only nonempty compact (weakly compact) convex invariant subset of $ {L_p}(G)\;({L_1}(G))$. We also characterize affine continuous mappings from $ {P_1}(G)$ into a bounded closed invariant subset of $ {L_p}(G)$ which commute with translations, where $ {P_1}(G)$ denotes the set of nonnegative functions in $ {L_1}(G)$ of norm one. Our results have a number of applications to multipliers from $ {L_q}(G)$ into $ {L_p}(G)$.


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DOI: https://doi.org/10.1090/S0002-9947-1977-0477604-5
Keywords: Locally compact group, left translations, right translations, convex sets, multipliers, operators commuting with translations, $ {L_p}$-spaces, measure algebra, convolution, multipliers, affine mappings
Article copyright: © Copyright 1977 American Mathematical Society